Connecting the Polynomial Remainder Theorem to Graphs

Miacademy & MiaPrep Learning Channel

This educational video explores the relationship between the Polynomial Remainder Theorem and the graphs of functions, specifically focusing on how to identify factors without performing long division. The narrator, Randy, begins by reviewing the definition of the theorem, which states that when a polynomial f(x) is divided by a linear expression (x-a), the remainder is equal to f(a). He demonstrates this with an algebraic example, showing how substituting a value into the function can quickly reveal the remainder. The video then transitions to a visual problem involving the graph of a polynomial function. Through a multiple-choice question, viewers learn how to deduce which linear expression results in a remainder of zero by identifying factors in the numerator. The narrator explains the logical connection: for a remainder to be zero, the divisor must be a factor of the dividend. Finally, the lesson connects these algebraic concepts to graphical features. It illustrates that a factor like (x-2) corresponds to an x-intercept (or zero) at x=2 on the graph. This video is highly useful for Algebra II and Pre-Calculus teachers to help students conceptualize the link between algebraic factors, remainders, and visual graphs, moving beyond rote calculation to deep conceptual understanding.